Domanda del libro di testo
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
49. lim (x → 0) (x - sin x) / (x tan x)
39
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.2.61
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
61. y = √(θ + 3) sin θ
Guida verificata passo dopo passo1
Start by expressing the function clearly: \(y = \sqrt{\theta + 3} \sin \theta\). Recognize that this is a product of two functions of \(\theta\): \(u(\theta) = \sqrt{\theta + 3}\) and \(v(\theta) = \sin \theta\).
Rewrite \(y\) in a form that is easier to differentiate logarithmically: \(y = (\theta + 3)^{\frac{1}{2}} \sin \theta\).
Take the natural logarithm of both sides to apply logarithmic differentiation: \(\ln y = \ln \left( (\theta + 3)^{\frac{1}{2}} \sin \theta \right)\).
Use logarithm properties to separate the terms: \(\ln y = \frac{1}{2} \ln (\theta + 3) + \ln (\sin \theta)\).
Differentiate both sides with respect to \(\theta\), remembering that \(\frac{d}{d\theta} (\ln y) = \frac{1}{y} \frac{dy}{d\theta}\), and apply the chain rule and derivative rules for logarithms and trigonometric functions.
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Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers of functions by taking the natural logarithm of both sides. This simplifies the differentiation process by converting multiplication into addition and powers into products, making it easier to apply derivative rules.
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06:30
Logarithmic Differentiation
Product Rule
The product rule is a fundamental differentiation rule used when finding the derivative of a product of two functions. It states that the derivative of f(θ)g(θ) is f'(θ)g(θ) + f(θ)g'(θ), which helps in differentiating expressions like y = √(θ + 3) sin θ.
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05:18The Product Rule
Chain Rule
The chain rule is used to differentiate composite functions, where one function is inside another. For example, when differentiating √(θ + 3), the chain rule helps by differentiating the outer function (square root) and multiplying by the derivative of the inner function (θ + 3).
Video consigliato:
05:02Intro to the Chain Rule
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